davie, c.t., pearce, c.j. , and bićanić, n. (2014) fully...

Davie, C.T., Pearce, C.J., and Bićanić, N. (2014) Fully coupled, hygro-thermo-mechanical sensitivity analysis of a pre-stressed concrete pressure vessel. Engineering Structures, 59 . pp. 536-551. ISSN 0141-0296 Copyright © 2014 The Authors A copy can be downloaded for personal non-commercial research or study, without prior permission or charge Content must not be changed in any way or reproduced in any format or medium without the formal permission of the copyright holder(s)

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Fully coupled, hygro-thermo-mechanical sensitivity analysis of apre-stressed concrete pressure vessel q

C.T. Davie a,⇑, C.J. Pearce b, N. Bicanic b

a School of Civil Engineering and Geosciences, Newcastle University, Newcastle upon Tyne, UKb School of Engineering, University of Glasgow, Glasgow, UK

a r t i c l e i n f o

Article history:Received 28 August 2012Revised 24 July 2013Accepted 22 October 2013Available online 15 December 2013

Keywords:Nuclear power plantConcreteHygro-thermo-mechanicalModelFinite element analysis

a b s t r a c t

Following a recent world wide resurgence in the desire to build and operate nuclear power stations as aresponse to rising energy demands and global plans to reduce carbon emissions, and in the light of recentevents such as those at the Fukushima Dai-ichi nuclear power plant in Japan, which have raised questionsof safety, this work has investigated the long term behaviour of concrete nuclear power plant structures.

A case example of a typical pre-stressed concrete pressure vessel (PCPV), generically similar to severalpresently in operation in the UK was considered and investigations were made with regard to theextended operation of existing plants beyond their originally planned for operational life spans, and withregard to the construction of new build plants.

Extensive analyses have been carried out using a fully coupled hygro-thermo-mechanical (HTM) modelfor concrete. Analyses were initially conducted to determine the current state of a typical PCPV after 33+years of operation. Parametric and sensitivity studies were then carried out to determine the influence ofcertain, less well characterised concrete material properties (porosity, moisture content, permeabilityand thermal conductivity). Further studies investigated the effects of changes to operational conditionsincluding planned and unplanned thermal events.

As well as demonstrating the capabilities and usefulness of the HTM model in the analysis of such prob-lems, it has been shown that an understanding of the long-term behaviour of these safety–critical struc-tures in response to variations in material properties and loading conditions is extremely important andthat further detailed analysis should be conducted in order to provide a rational assessment for life exten-sion.

It was shown that changes to the operating procedures led to only minor changes in the behaviour ofthe structure over its life time, but that unplanned thermal excursions, like those seen at the FukushimaDai-ichi plant could have more significant effects on the concrete structures.

� 2013 The Authors. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Until recently, in many parts of the world, such as in the UK andother parts of Europe nuclear energy has fallen out of favour. But,with an ever increasing world demand for energy and the threatof global warming, nuclear power is now seen again by many asa reliable, plentiful and most importantly low carbon supply ofelectricity [1,2]. As a result of this, there is currently a worldwideresurgence in the development and application of nuclear power

with several countries including China, India and the UK recentlyapproving the development of new build plants [1].

However, while the international treaties to reduce carbonemissions must be enacted in relatively short timescales [3,4] theprevious stagnation of the nuclear industry has resulted in a leadin time of many years before new nuclear plants can be commis-sioned [1]. To fill this gap, the lives of the existing stock of nuclearpower plants in the UK and Europe, many of which are reachingthe end of their original design lives, will need to be extended.

At the same time, the incidents following the Great East Japanearthquake in 2011, where a loss of cooling at the FukushimaDai-ichi plant led to the overheating of several reactors and the re-lease of radioactive material, have again raised questions as to thesafety of nuclear power plants [5]. The main line of defence arounda reactor to prevent escape of radioactive material is the contain-ment vessel. To ensure that there is not a repeat of some of the pastnuclear accidents, it is important that structural integrity of the

0141-0296/$ - see front matter � 2013 The Authors. Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2013.10.033

q This is an open-access article distributed under the terms of the CreativeCommons Attribution-NonCommercial-No Derivative Works License, which per-mits non-commercial use, distribution, and reproduction in any medium, providedthe original author and source are credited.⇑ Corresponding author. Address: School of Civil Engineering and Geosciences,

Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom. Tel.: +44(0)191 222 6458.

E-mail address: [email protected] (C.T. Davie).

Engineering Structures 59 (2014) 536–551

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containment vessel is maintained under all conceivable circum-stances and conditions. To achieve this, it is essential that not onlytheir mechanical properties are understood but, as they are sub-jected to extreme environments, also how these change over theoperational life of the reactor vessel.

This paper presents an assessment of the long term behaviourof pre-stressed concrete pressure vessels (PCPV), typical of designscurrently employed in the UK, that have been operating at elevatedtemperatures for periods in excess of 30 years. This assessment isimportant as it evaluates the current mechanical properties ofthe concrete which may then be used to ensure the structuralintegrity and containment of facilities whose life is to be extended.

The current state of these vessels cannot be identified withoutaccounting for the full history of the operating conditions experi-enced by the structure and consideration of the effects of thermaland mechanical loads on the concrete including transport of mois-ture and the development of gas pressures within the pore struc-ture of the material.

To achieve this, a fully coupled hygro-thermo-mechanical(HTM) model for concrete, originally developed during the EUFP5 Euratom MAECENAS (Modelling of Ageing in Concrete NuclearPower Plant Structures) project, was employed to examine thebehaviour of a typical UK PCPV over the course of its 30+ year lifespan under various operating conditions.

In the first instance analysis of a typical PCPV is carried out todetermine the current condition of the concrete structure. A para-metric analysis is then presented in which consideration is given tovariations in the concrete with respect to less well characterisedproperties such as porosity, moisture content, permeability andthermal conductivity. Although the records from the original de-sign and construction of UK PCPV plants are well documentedand maintained, little is known about some of these propertiesand examination of the literature shows that very wide rangesmay exist (see for example Fig. 1).

This work demonstrates that the long term behaviour of thesesafety–critical structures is very susceptible to small changes inmaterial properties and a good understanding of this sensitivityis therefore vital.

Finally, a parametric study is presented in which the influenceof operational conditions on the mechanical properties of the PCPVis demonstrated. This study shows how planned or unplannedevents may affect the overall structural behaviour and how thismay affect the safety of the PCPV. It is furthermore concluded thatthe numerical tool applied in this paper may be used to predict thebehaviours of existing or new build concrete power plant struc-tures (including but not limited to PCPV).1

2. Numerical model

This work was carried out using the fully coupled hygro-ther-mo-mechanical (HTM) model for concrete initially developed dur-ing the MAECENAS (Modelling of Ageing in Concrete Nuclear PowerPlant Structures) project and presented by Davie et al. [6]. Briefly,the model treats concrete as a multiphase porous medium consist-ing of solid, liquid and gas phases. The solid skeleton is consideredto behave isotropically and elastically under mechanical and ther-mal loadings, although nonlinear responses are accounted forthrough the consideration of transient thermal strains, transientthermal creep and an isotropic thermo-mechanical damage formu-

lation. The ‘liquid’ phase considers free liquid water in pores andadsorbed water physically bound to the surface of solid skeleton.Water liberated from the solid skeleton through dehydration isconsidered as a part of free liquid water since chemically boundwater is assumed to be initially released as liquid water. The gasphase is considered to be a mixture of dry air and water vapour,which are assumed to behave as ideal gases. Most of the materialproperties, both mechanical and those related to heat and masstransport, are variable, often either directly or indirectly, as a func-tion of temperature. A complete description of the governing equa-tions, constitutive laws and material properties may be found in[6,7,17]. A brief outline of the components of the formulation keyto this work is given below and in Appendix AI.

2.1. Governing equations

The model comprises four conservation equations for mass ofdry air (1), mass of moisture (i.e., vapour and liquid) (2), energy(3) and linear momentum (4).

@ðeG ~qAÞ@t

¼ �r � JA ð1Þ

@ðeG ~qV Þ@t

þ @ðeLqLÞ@t

� @ðeDqLÞ@t

¼ �r � ðJV þ JLÞ ð2Þ

ðqCÞ @T@t� kE

@ðeLqLÞ@t

þ ðkD þ kEÞ@ðeDqLÞ@t

¼ r � ðkrTÞ þ kEr � JL ð3Þ

r � ðr0 � gPPoreIÞ þ b ¼ 0 ð4Þ

where eh is the volume fraction of a phase h(h = L, V, A, G, D refer toliquid water, water vapour, dry air, gas mixture and dehydratedwater phases, respectively), qh is the density of a phase h, ~qh themass of a phase h per unit volume of gaseous material, Jh the massflux of a phase h, qC the heat capacity of concrete, k the effectivethermal conductivity of concrete, kE and kD are the specific heatsof evaporation and dehydration, r0 is the Bishop’s stress (alsoknown as the effective stress in geomechanics), I the identity ma-trix, g is the Biot coefficient, PPore the pore pressure and b the bodyforce.

2.2. Fluid transport equations

Liquid water flow in the pore structure of the concrete isassumed to be driven by pressure according to Darcy’s law, andgas flow is assumed to be driven by both pressure and concentra-tion according to Fick’s law. The mass fluxes of dry air (JA), water

Fig. 1. Values for porosity vs. permeability for ordinary concretes (OC) and highperformance concretes (HPC) described in the literature ([6–16]).

1 While the specific example considered here is of a PCPV, the findings of the workclearly have implications for other high temperature concrete structures includingbut not limited to pre-stressed concrete containment vessels (PCCV). Furthermore,while the initial example considered here considers a typical existing structure, thereare clear implications for assessing the long term behaviour of new build projects,under desirable and undesirable conditions.

C.T. Davie et al. / Engineering Structures 59 (2014) 536–551 537


vapour (JV) and liquid water (JL) per unit area of concrete are givenby Eqs. (5)–(7).

JA ¼ eG ~qA �kgKKG

lG

� �rPG � eG ~qGDAVr

~qA

~qG

� �ð5Þ

JV ¼ eG ~qV �kgKKG

lG

� �rPG � eG ~qGDAVr

~qV

~qG

� �ð6Þ

JL ¼ eLqL �KKL

lL

� �rPL ð7Þ

where K is the intrinsic permeability of the dry concrete, Kh, lh andPh are the relative permeability, dynamic viscosity and pressure ofthe phase h, kg is the gas-slip modification factor and DAV is the coef-ficient of diffusion for the dry air/water vapour mixture within theporous concrete.

The moisture content and saturation of the concrete is con-trolled by the relative humidity in the pore space via the sorptionisotherms (8).

eL ¼eCemqCem

qL� f PV

PSat; T

� �ð8Þ

where eCemqCem is the cement content per unit volume of concrete,PV is the vapour pressure, PSat is the saturation vapour pressure and(PV /PSat) is the relative humidity (see AI.3).

The porosity, /, of the concrete then controls the size of the gasvolume fraction, eG (9).

/ ¼ eL þ eG ð9Þ

2.3. Mechanical constitutive equations

The total strain (e) of the solid skeleton is considered to consistof elastic strain (ee), free thermal strain (eft) and load induced ther-mal strain (elits), i.e.,

e ¼ ee þ eft þ elits ð10Þ

The effect of mechanical and thermal loading on the elastic stiffnessis accounted for by an isotropic scalar damage model where theclassical damage formulation, with a single mechanical damageparameter, x, has been modified to include a second thermal dam-age parameter, v.

r0 ¼ ð1�xÞð1� vÞD0 : ee ð11Þ

where r0 is the Bishop’s stress, D0 is the initial elasticity tensor, x isthe mechanical damage parameter, accounting for the loss of theelastic stiffness caused by the micro-fracturing of concrete thatdevelops under loading and v the thermal damage parameter,accounting for the reduction of the elastic stiffness due to thermallyinduced degradation of the cement paste.

The free thermal strain rate is calculated by way of Eq. (12):

_eftij ¼ a _Tdij ð12Þ

where a is a non-linear, temperature dependent coefficient of ther-mal expansion, _T is the rate of temperature change and dij is theKronecker delta.

The load induced thermal strain rate is calculated as shown inEq. (13):

_elitsij ¼

bf 0cð1� mcÞr0�ij � mcr0�kkdij

� �_T for _T > 0 ð13Þ

where b is the coefficient of load induced thermal strain (AI.5), f 0c is

the initial compressive strength, mc is the lateral component of theload induced thermal strain (similar to Poisson’s effect) and r0�ij is

the negative (compressive) projection of the Bishop’s stress tensor,r0ij.

2.4. Numerical solution

Using the standard Finite Element approximation, the chosenprimary variables of displacements, u, temperature, T, gas pressure,PG, and vapour content, ~qV , are expressed in terms of their nodalquantities:

u ¼ Nua; T ¼ NT T; PG ¼ NPPG; ~qV ¼ NqqV ð14Þ

where Nh are the shape functions, a, T, PG and qV are the nodal vari-ables, and the resulting system of discrete equations is expressed inmatrix form as:

C _xþ Kx ¼ fext ð15Þ

The discrete set of Eq. (15) is also discretised in time using a finitedifference scheme as:

xtþaDt ¼ ð1� aÞxt þ axtþDt _xtþaDt ¼ xtþaDt � xt

Dtð16Þ

where Dt is the time increment, xt and xt+Dt denote the unknownsat times t and t + Dt respectively and a is a constant (0 6 a 6 1).

Boundary conditions are defined for the primary variables aseither Cauchy type (mixed) boundary conditions, where energyand mass transport takes place across the boundary dependenton the conditions internal and external to the concrete, or Dirichlettype (fixed) boundary conditions, depending on the nature of thephysical boundaries themselves.

2.5. Model validation

This model has been validated through extensive application tonumerous problems ranging from isothermal drying to rapid fireloading, and has been demonstrated capable of accuratelyrepresenting the multi-phase, macro-scopic behaviour of concreteexposed to elevated temperatures, e.g. [6,7,17].

3. Numerical analyses

The results from a total of 32 analyses are presented covering arange of parametric investigations, as described in the followingsections. The scale and complexity of the PCPV structure, whichis typical of those still in use in several plants in the UK, is illus-trated by the schematic shown in Fig. 2 and finite element meshesshown in Fig. 3. The analyses conducted for this work are of twotypes. The first type employs the axi-symmetric representation ofthe PCPV (Fig. 3b) to conduct full HTM analyses of the vessel.The second type employs the simplified axi-symmetric representa-tion of a slice through the outer wall of the vessel (Fig. 3c) in orderto conduct a parametric sensitivity study of the properties affectingheat and mass transport in the concrete.

In all cases the boundary conditions for the analyses are set upsuch that the outside of the pressure vessel is considered to beopen to the atmosphere and can thus freely exchange vapour andheat (see Fig. 3). Typical values of atmospheric properties aremaintained with a constant pressure of 0.1 MPa and temperatureof 15 �C. For all but one analysis, the relative humidity of the atmo-sphere is set to 70%.

The inside of the pressure vessel is considered to be sealed togas flow, as representative of the internal steel liner, and has ap-plied a set of prescribed temperatures, which vary in time accord-ing to the operational condition of the reactor, as exemplified inFig. 4c. For the full HTM analyses the typical magnitude of the tem-peratures vary depending on location around the inside surface,

538 C.T. Davie et al. / Engineering Structures 59 (2014) 536–551


from a minimum operating temperature of 37 �C to a maximum of85 �C. For the simplified ‘slice’ analyses a single temperature pro-file is considered.

For the full axi-symmetric analysis three types of mechanicalboundary condition were also required. Firstly, the base of the typ-ical structure rests on a stiff but deformable neoprene layer repre-sented by elastic spring boundaries along the inner part of thelower edge of the mesh. Secondly, to represent the pre-stressingloads provided by helical tendons tensioned throughout the out-side wall of the structure equivalent vertical and horizontal nodalloads were applied to the top and bottom boundaries as well asinternal elements. The third set of mechanical boundary conditionswere nodal forces applied to the internal surface of the vessel torepresent the internal pressures acting on the structure duringoperation.

Typical variations of the pre-stressing forces over time, account-ing for creep in the tendons, and of the internal pressures, depen-dent on the operational conditions of the reactor over its life time,are shown in Fig. 4.

The details of the typical boundary condition data was providedby the operators during the MAECENAS project from records of con-struction and operation [18]. Specific details of the applied bound-aries are discussed as necessary in the following sections.

4. Numerical investigations

4.1. HTM analysis of PCPV

This first analysis was conducted in order to determine the cur-rent state of a typical PCPV structure resulting from its 30+ year

operating life time. As described in the previous section a fullHTM analysis was carried out using the axi-symmetric meshshown in Fig. 3b and applying representative temperature, pres-sure and load histories as recorded over the life time of the struc-ture (Fig. 4).

As can be seen from Fig. 4 the first approximately 7 years of thestructures life were considered the construction phase duringwhich the reactor was not commissioned and no increased temper-atures were experienced by the structure. However, during theconstruction phase, after approximately 4½ years and followingtensioning of the pre-stressing tendons, a proof pressure test wasconducted during which the internal pressure was raised to nearly5 MPa, well in excess of the typical operating pressures of 3.9 MPa.Following the construction phase and reactor start up, it may beseen that the temperatures and pressures on the internal faces ofthe vessel increased smoothly to their operational values and thenremained constant for the life time of the structure, with theexception of two planned outages after approximately 9 yearsand 22 years of operation.

For this analysis a best estimation of the initial materialproperties was employed, taking what data was available fromthe operator’s construction and maintenance records for the typi-cal structure. The initial material properties employed are listedin Table 1 [18].

The results of the analysis can be seen in Figs. 5–7. Fig. 5a showsthe temperature profile in the concrete structure. This profile, ta-ken at the end of the modelled life time is typical of the near steadystate conditions found to develop during long term operation. Itcan be seen that the temperature on the inside surface varies withlocation, the hottest parts being at the top of the vessel. It can alsobe seen that the temperature field extends most of the waythrough the �5 m thick concrete structure and so, although at rel-atively low levels, much of the concrete is subject to effects includ-ing thermal expansion, transient thermal creep and thermaldamage (i.e. degradation of the concrete properties and the devel-opment of micro-cracking as a result of heating).

Fig. 5b shows the gas pressure field in the concrete structurewhere pressures have developed inside the pore space of the por-ous concrete material as a result of the evaporation of liquid waterand the subsequent development of water vapour. As can be seenand expected, this field corresponds with that of the temperatures,with the highest pressures developed in the hottest areas of theconcrete, nearest the inner steel liner and in the upper parts ofthe vessel. While the maximum pressures predicted here are ofthe order of 0.29 MPa and pose few implications for the structure,it is notable that under relative mild conditions pressures of nearly

Reactor core

30m

35 m

Pre-stressed Concrete Pressure Vessel

Steel Liner

Cooling pipe and boiler feed

penetrations etc.

Stand pipes

Fig. 2. Schematic illustration of typical PCPV.

Fig. 3. (a) Quarter symmetry 3-dimensional finite element mesh of typical PCPV, (b) simplified axi-symmetric finite element mesh of typical PCPV, (c) axi-symmetric finiteelement mesh of slice through outer wall of typical PCPV.



3 times atmospheric pressure developed close to the steel linerthat provides containment and that most of the structure is subjectto elevated pressures. The development of these pressures and po-tential implications will be considered further in the followinganalyses.

Corresponding with the gas pressure field, the saturation of thepressure vessel with liquid water may be seen in Fig. 5c. It may benoted that a significant proportion of the vessel is subject to dryingfrom the initial �75% saturation to a minimum of �40% at the hot-test parts. Nonetheless, all of the concrete remains wet to some ex-tent and in fact some of the deeper part of the vessel experience aslight increase in saturation (to a maximum of �77%) as moisturemoves from the hotter parts of the vessel and re-condenses in thecooler parts. It may also be seen that the outer most parts of theconcrete experience a slight drop in saturation as drying to theatmosphere takes place over time.

The mechanical response of the structure over the modelled lifetime is illustrated in Fig. 5d–g. The deformation of the structuremay be noted in Fig. 5d–f and, although exaggerated in these fig-ures, it can be seen that the shape of the structure changes consid-erably during the cold and hot operational phases. During the coldphases, pre-operation (Fig. 5d) and at the end of the second outage(Fig. 5e) it may be seen that the wall of the structure bends inwardsunder the confining loads of the pre-stressing tendons. However,during the hot phases, the wall of the structure bends outwards,under the combined actions of restrained thermal expansion andinternally applied pressures.

Relatively, the deflections are not large, reaching a maximumlateral displacement at the end of the modelled life time of approx-imately 1.3 mm, at mid-height on the outside of the wall of thestructure. However, in considering the size and stiffness of thestructure this may be considered noteworthy.

It may be further noted that the lateral deformation increasesover the life time of the structure. This is illustrated by Fig. 5e,where it can be seen that, although the structure has returned toan inwardly bending configuration during the (cold) outage, thepattern of deformations is different to that seen at the end of theconstruction phase (Fig. 5d). This is further illustrated by Fig. 6,which shows the lateral deformation in time at mid-height onthe outside of the wall of the structure.

As can be seen, a large inward (negative) deformation occursinitially when the confining pre-stress loads are applied duringthe construction phase. An outward (positive) spike occurs dueto the proof pressure test after approximately 5 years and this isthen followed by a series of large outwards and inwards displace-ments that are associated with the bending behaviour of the struc-ture during initial heat up of the reactor and the two plannedoutages. However, overlying all of these, a gradual outwards in-crease in deformation with time may be seen. This is due firstlyto the creep of the pre-stressing tendons which gradually relax

(a)

(b)

(c)

Fig. 4. (a) Load factor applied to pre-stressing tendon loads over life time of PCPV,(b) Internal pressures applied to internal surface of PCPV during operation over itslife time, (c) Typical temperature profile applied to internal surface of PCPV duringoperation over its life time.

Table 1Initial concrete properties and boundary conditions.

Property Value Boundary condition Value

Tensile strength 4 MPa InternalBoundary

Temp. Prescribed

Compressive strength 60 MPa Pressure Undefined (sealed conditions)Porosity 0.10 Vapour Undefined (sealed conditions)Permeability 2.0 � 10�18 m2 Mechanical Applied loads according to pressure profile (Figs. 3b and 4b)

Mass moisture content/saturation

3.3%/�75% ExternalBoundary

Temp. 15 �C

Thermal conductivity Average Eurocode curve [19](AI.2)

Pressure 0.1 MPa

Density 2270 kg/m3 Vapour 0.00898 kg/m3 � 70% Relative HumidityMechanical Applied loads according to pre-stressing load profile (Figs. 3b

and 4a)Spring boundaries along base (Fig. 3b)



the confining loads (Fig. 5a), and secondly, to the development ofmechanical damage, which can be seen in Fig. 5g.

The damage occurs when the wall of the structure bends out-wards during the hot operational phases, resulting in the develop-ment of tensile stresses, and forms in a zone along the outside faceof the structure and deeper into the structure at mid-height. Whilethis damage may not represent an observable phenomenon interms of visual inspection of the pressure vessel, it does indicate

that the material in this area, and therefore the structure, hasdegraded from its original strength, stiffness and structuralintegrity; the reduction in stiffness leading to the increase indeformation.

Upon further examination it may be noted that the level ofdamage itself increases over the life time of the structure. Fig. 7shows the development of damage in time at 6 points along thelength of the damage zone seen in Fig. 5g.

Fig. 5. Deformation of structure at an exaggerated scale (�500) showing (a) temperature [K] field under long term operational conditions, (b) pore gas pressure [Pa] fieldunder long term operational conditions, (c) liquid water saturation field under long term operational conditions, (d) lateral displacements [m] under cold conditions, preinitial heat up phase, (e) lateral displacements [m] at end of second outage, (f) lateral displacements [m] under long term operational conditions, (g) mechanical damage fieldunder long term operational conditions.

Fig. 6. Lateral displacements at mid-height on the outside of the wall of thestructure.

Fig. 7. Mechanical damage development (see Fig. 5g) at points measured from theoutside surface of the structure, in time.



As can be seen damage originally develops at the outside of thestructure (0.0 m) upon first heat-up (see Fig. 4c) as a result of nor-mal operating conditions. Damage then begins and continues todevelop at other points deeper in the structure, sometimes, butnot always, concurrent with restart of the reactor after the twoplanned outages. This progressive development further explainsthe increase in lateral deformations noted earlier, which developas a result of the progressive decrease in structural stiffness.

Overall, it is likely that the growth in damage is due to the grad-ual decrease in the confining loads resulting from the creep in thepre-stressing tendons combined with the effects of transient ther-mal creep (which, due to its monotonic occurrence, can lead to thedevelopment and ‘locking in’ of tensile stresses during cool down[20]). The deepest point at which damage develops is approxi-mately 1 m from the outside surface of the vessel. This is reason-ably significant in terms of the overall size of the structure butthe damage is likely to be micro-structural rather and observablemacro-scale fractures and furthermore, the deflections remain verysmall. So the results do not suggest an imminent loss of structuralintegrity or a particular safety concern. However, they do suggestthat detailed analysis is required over the life time of the structurein order to provide a rational assessment for life extension. It canalso be seen that in terms of new build, this type of analysis maybe employed very usefully to inform design decisions such as thetype and magnitude of pre-stress loads.

4.2. Simplified slice analysis – parametric study

As described previously, following the full analysis described inSection 4.1, a set of 22 analyses were carried out in order, firstly, toconduct a parametric investigation of the effects of various, lesswell identified material properties; porosity, moisture content,permeability, and thermal conductivity, and secondly, to conducta sensitivity study in relation to operating conditions experiencedby the structure; timing and length of shut downs, environmentalconditions and extended operating life. These investigationsfocused on the transient heat and mass transport in the structureand so a simplified slice model was employed, as shown inFig. 3c. Mechanical behaviour was not specifically considered. Asbefore, unless specified as part of the analysis, the typical recordedtemperature history was applied to the inner face of the structureover the full 30+ year operating life time, and again, unless specificto the analysis all initial material properties remained the same asthose applied in the first analysis (Table 1). Of principal concernduring these analyses were the temperature profiles across thestructure and the gas pressures built up in the pore space of theconcrete at the internal boundary. The temperature affects mate-rial properties such as strength and stiffness, which degrade withincreasing temperatures, transient strains (both thermal expansionand transient thermal creep), and the development of damage. Thegas pressures effectively apply a tensile stress internally to theconcrete and therefore have implications for the development ofdamage. At the internal boundary of the concrete structure it isalso possible that these pressures will act directly on the steel liner.

4.2.1. PorosityPorosity is the space available in the pore structure of the con-

crete in which fluids (liquid and gas) can be present and throughwhich they can flow. It therefore controls fluid pressures and flowbehaviour and so may have a significant bearing on the overallcouple HTM behaviour of a structure such as a PCPV.

Unfortunately porosity it is not usually specified in design oreven measured for in existing structures, so its value is not gener-ally known. Examination of the literature finds numerous valuesfor the porosity of concrete quoted ranging from around 5% to

around 16%. These include values for high performance and ordin-ary concrete with a wider range being seen for the latter (see Fig. 1).

However, little pattern or trend is discernible in the quoted val-ues and it is difficult to directly relate the porosities to any otherfactor or property of the concretes. Without laboratory measure-ments it is therefore very difficult to identify the value for a givenconcrete.

Although the value of 10% employed in the first analysis was abest estimate from the information available, it is far from certainthat this is exactly correct. Therefore, to determine the significanceof the porosity for the behaviour of the PCPV three separate anal-yses were conducted with porosities of 5%, 10% and 16%.

Fig. 8a shows the near steady state temperature profile devel-oped across the width of the PCPV wall during periods of normaloperating conditions. As can be seen, although it may be expectedthat variations in the moisture content and transport through thewall, as may be caused by variations in porosity, could affect theheat transfer behaviour (and vice versa) through coupling of prop-erties such as the specific heat capacity, virtually no effect wasseen in this case. So it may be stated that the porosity has no sig-nificant effect on the heat transfer and this was in fact found to bethe case for all properties investigated here.

In relation to the moisture transport, however, there are twopoints of note. Firstly, it may be seen that the gas pressures atthe inner boundary of the structure continued to develop andchange over the life time of the structure (Fig. 8b). Clear changesin the gas pressure can be seen that are easily identifiable withthe relatively rapid changes in temperature associated with firstheat up and the two planned outages. However, the size of thestructure and the relatively low operating temperatures mean thatthe drying process is extremely slow and is not complete even after30+ years of operation (Fig. 8c). Relatively, the lower porositymaterials dry faster, where an initial saturation of 75% representsa smaller total volume of water, and it might be expected thatthe drying rate would therefore be proportional to the porosity.However, the drying rate is also affected by capillary suction,which acts to hold the water in place. This is lost below a satura-tion of 0.55, which is representative of the point of maximum cur-vature of the capillary menisci and the point at which they can nolonger be supported. As can be seen in Fig. 8c, when capillary suc-tion is lost, the drying rate then increases.

The second point of note is that the porosity of the concrete hasa reasonably significant effect on gas pressure build up. With theoriginal porosity of 10% the gas pressure peaked after each heatup phase and reached an overall peak after approximately25 years, after which it slowly decreased. The maximum pressurereached was just under 0.3 MPa. With the lower porosity of 5%,where there is less space and therefore less water available forevaporation, the gas pressure decreased slowly over the whole lifetime of the structure and reached a maximum pressure of less than0.2 MPa. However, with the higher porosity of 16%, where mostwater is available for evaporation, it can be seen that the gas pres-sure continued to rise over the full life time of the structure. Themaximum pressure reached after approximately 33 years ap-proached 0.6 MPa. It should be noted that while there may be atangible link between porosity and permeability, here permeabilityremained constant and variations in permeability for constant val-ues of porosity are considered later.

So, for the feasible range of porosities found in the literature itmay be seen that the gas pressure ranges up to 6 times atmo-spheric pressure. While the magnitudes in question are not inthemselves necessarily significant at this point in terms of theloads applied to the concrete or steel liner, it may been seen thatfor higher porosities the conditions worsen over the life time ofthe structure and the observed variation is significant enough soas to suggest that a reasonable understanding of the concrete



porosity is required in order to gain a reasonable prediction of themoisture transport in the concrete.

4.2.2. Moisture contentsAs mentioned in relation to the porosity, the gas pressures that

can develop in the concrete are directly related to the amount ofwater available for evaporation. They are therefore fundamentallyrelated to the initial moisture content of the concrete, which is afunction of the concrete mix and of the atmospheric conditionsduring curing, with which the water in the porous concrete willequilibrate. Furthermore, the amount of water present in the con-crete will affect both the thermal conductivity and the heat capac-ity of the concrete and will therefore affect the heat transfer. Since,as discussed previously, the development of gas pressures in thepore space of the concrete could lead to the development of dam-age, it may be important to consider the precise moisture contentof the concrete.

Again, however, the true moisture content is not easily estab-lished due to its dependence on the original design mix and theenvironmental conditions subsequent to construction, and so todetermine its significance a range of moistures have been investi-gated. Three analyses were run using the simplified slice modelwith initial saturations of 65%, 75% and 85% (approximately equiv-alent to mass moisture contents of 2.9%, 3.3% and 3.8%).

From Fig. 9a it can be seen that the gas pressures were alsoinfluenced by the initial moisture content although the range andeffect was much less than that caused by the variation in porosity,with a maximum pressure of about 0.35 MPa predicted. Moisture

content may therefore be considered to be of low significancewhen considering the structural performance of the PCPV.

Notwithstanding this conclusion, it may be seen that while formass moisture contents of 3.3% and below the gas pressurespeaked before reducing slowly over the life time of the structure,for a mass moisture content of 3.8% the gas pressures continuedto rise over the whole life time of the structure and therefore rep-resented a worsening condition.

The initial moisture content also affected the drying response ofthe structure (Fig. 9b). Again, the drying rates increased for thelower initial moisture contents as the saturation dropped below0.55 and in these cases the drying process was not complete evenafter 30+ years of operation. In contrast, with the highest initialmoisture content the saturation remained above this critical pointand levelled off at a steady state, implying equilibrium with theatmosphere, only a few years after each outage. This higher ‘final’moisture content could have significance for structural perfor-mance in the light of operational conditions or temperature excur-sions experienced in later life (see Sections 4.3 and 4.4).

4.2.3. PermeabilityWhile laboratory testing is desirable to determine the porosity

and moisture content of a concrete, estimates for these propertiesmay be made if the original mix design is known and assumptionsare made about the degree of hydration. However, no such esti-mate may be made as to the permeability of the concrete as, whilethis can be linked to the porosity, it is controlled by the microstruc-ture of the concrete and this is not easily related to mix design.

Fig. 8. Results of analyses with various porosities showing (a) steady state temperature profiles across the width of the structure, (b) gas pressures at the inner boundary intime and (c) the liquid water saturation at the inner boundary in time.



Concretes of similar porosity may have different permeabilitiesand laboratory testing is therefore the only realistic way of deter-mining a value.

As with porosity, a very wide range of values for permeabilityare quoted in the literature ranging across six orders of magnitudefrom 10�16 m2 to 10�21 m2 for ordinary concretes and from10�18 m2 to 10�22 m2 for high performance concretes (see Fig. 1).Again, there is little identifiable pattern to the quoted values otherthan, as would be expected for a denser material, HPC tends tohave lower predicted values (but still ranging over four orders ofmagnitude). This again means that, from an analysis point of view,it is difficult to choose an appropriate value for a particularconcrete.

Therefore, to determine the significance of the permeability forthe behaviour of the PCPV three separate analyses were conductedwith permeabilities of 1.0 � 10�16 m2, 2.0 � 10�18 m2 and1.5 � 10�21 m2.

From the results shown in Fig. 10a it can be seen that the per-meability had a significant effect on the predicted gas pressures.As discussed previously, for the original permeability the gas pres-sures peaked at around 0.3 MPa after about 25 years before slowlydecreasing. With the highest permeability, gas pressures reachedlittle more than 0.15 MPa, peaking after each heat-up phase before

dissipating quickly to an almost steady state condition. With thelowest permeability the gas pressures continued to rise over thelife time of the structure, and as has been discussed before this isa persistently worsening situation. However, the peak pressure,reached after about 33 years, is around 1.05 MPa. This pressure issignificant, not just because of the very large effect it shows thepermeability to have of the gas pressure behaviour, but also be-cause the magnitude of the pressure is of an order similar to thetensile strength of the concrete. If these conditions prevailed then,in contribution with other (mechanical) stresses, it is conceivablethat damage could be incurred in the concrete near the steel liner.The load acting on the liner itself is also perhaps of concern.

In considering the moisture content (Fig. 10b) it can be seenthat for the lowest permeability concrete, the drying process isvery slow and very little moisture is lost even after 30+ years ofoperation. In fact it can be seen that the moisture content increasedslightly upon first heat up. This is a result of a change in the equi-librium conditions within the concrete as vapour, formed fromevaporating liquid water, is restricted from moving away fromthe heated part of the structure.

In contrast it may be seen that the highest permeability con-crete dries relatively very quickly and appears to reach a steadystate a few years after each outage. It may be interesting to note

Fig. 9. Results of analyses with various moisture contents showing (a) gas pressures at the inner boundary in time and (b) the liquid water saturation at the inner boundary intime.

Fig. 10. Results of analyses with various permeabilities showing (a) gas pressures at the inner boundary in time and (b) the liquid water saturation at the inner boundary intime.



however that it also recovers moisture content much more quicklyduring cool downs than lower permeability concretes as vapour isable to move readily back into the cooling concrete.

4.2.4. Thermal conductivityThe thermal conductivity of the concrete is also a property

about which little can be determined without experimental test-ing. Thermal conductivity controls the rate at which heat transfersthrough the concrete and so indirectly affects the fluid transportand the development of gas pressures. Upper and lower boundtemperature dependent curves, considered to represent a rangeappropriate for all concretes (ordinary and high strength), are pre-sented in the Eurocode [19]. Although no guidance is given as tospecifically when to use what level of curve, it is suggested thathigh strength concretes may have higher thermal conductivitiesthan ordinary concretes but the UK National Annex states thatthe lower bound curve should be employed.

To determine the significance of the thermal conductivity forthe behaviour of the PCPV three separate analyses were conductedusing the upper, mid-range and lower bound Eurocode curves.

As can be seen from Fig. 11a, over the life time of the PCPV, thegas pressures at the inside surface were very slightly higher withlower thermal conductivities. This is a result of the slower dissipa-tion of the heat from the inside face through the wall of the struc-ture and the consequent increase in conversion of liquid water tovapour. This in turn resulted in slightly higher relative humidity,leading to a change in the equilibrium conditions and ultimatelya slightly higher saturation (Fig. 11b).

However, it may be noted that the effects were minimal for thisset of conditions, which involves relatively low temperatures act-ing over relatively long time periods, whereby near steady stateconditions are achieved. Hence, thermal conductivities within theranges suggested by the Eurocode may be considered of very lowsignificance in terms of the structural behaviour of the PCPV undernormal operating conditions.

Close examination of the temperatures in the structure overtime show that the near steady state temperature profiles acrossthe thickness of the wall were also minimally affected by the ther-mal conductivity with the temperature on the outside of the struc-ture varying by less than one Kelvin from those shown in Fig. 8a.The temperatures within the wall of the structure can be seen toincreasingly lag those at the inside surface with lower thermal con-ductivities, as would be expected. But again, the overall effectswere minimal and for all analyses near steady state conditions

were reached in a relatively short timescale relative to the life timeof the PCPV.

4.2.5. Worst case conditionsWhile some of the factors investigated above have been shown

to have minimal significance for the overall heat and mass trans-port behaviour in the PCPV several have been shown to representa worsening condition over the life time of the PCPV in relationto a continuous build up of gas pressure (stress) within the con-crete of the structure. For the ranges of property values that havebeen determined to be reasonable, and potentially unknown, fora given PCPV or similar concrete structure, a ‘worst case’ combina-tion may be imagined whereby all of the most adverse conditionsare found to be present.

To determine the significance of this ‘worst case’ combinationan analysis was run, following the results of the previous four setsof analyses, with a porosity of 12.2%,2 liquid water saturation of 95%saturation, a permeability of 3.0 � 10�21 m2 and a thermal conduc-tivity according to the lower bound curve of the Eurocode.

As can be seen from the results shown in Fig. 12a, the ‘worstcase’ combination resulted in gas pressures of approximately1.35 MPa, which, as was expected, were the highest yet predicted.Again, this pressure, although alone not a concern for the develop-ment of damage, is significant enough to warrant further consider-ation being as it is, of an order of magnitude comparable with thetensile strength of the concrete and continuing to rise over the lifetime of the structure. There is reason for concern that these pres-sures could, in time, contribute to the development of damage inthe concrete, in combination with other stresses, and that the steelliner could be affected.

It may also be noted that the moisture content of the concreteremained very high throughout its life time (Fig. 12b).

4.3. Simplified slice analysis – operating conditions

The following section is concerned with the sensitivity of thebehaviour of the PCPV to changes in operating conditions. As canbe seen from the typical profiles shown in Fig. 4, the PCPV experi-enced two planned outages during its life time and during these

Fig. 11. Results of analyses with various thermal conductivities showing (a) gas pressures at the inner boundary in time and (b) the liquid water saturation at the innerboundary in time.

2 While the ‘worst’ porosity considered in Section 4.2.1. was 16% and the ‘worst’permeability considered in Section 4.2.3 was 1.5e�21 m2, it is not realistic to considera concrete with extremes of both high porosity and low permeability. To give a morerealistic ‘worst case’, values for a concrete presented by Baroghel-Bouny et al. [8] havebeen used.



periods the temperatures and internally applied pressure returnedto normal, before increasing again upon reactor restart. Clearlythese changes affect the heat and moisture transport in the struc-ture of the PCPV and so, in order to identify implications for oper-ational and management strategies, a series of analyses wereconducted to determine the significance of various hypotheticalchanges to the operating conditions experienced by the structureduring its life time. The factors investigated were the timings andlength of the outages, the environmental conditions outside thestructure and the extended operation of the PCPV. In each case re-sults are presented (see Fig. 13a–n) using the typical propertiesand conditions, as employed above for the first analysis (Table 1),and using the ‘worst case’ conditions found in the previous section(Fig. 12a and b).

4.3.1. Timing of outagesComparing the gas pressures predicted under the typical regime

(Fig. 3) with earlier and later outages, and no outages at all, it canbe seen in Fig. 13a that under typical conditions the predicted gaspressure, after the full life span of the PCPV, varied very little. Aftereach outage the gas pressure peaked slightly before returningslowly to steady conditions. As a consequence of never peakingthe predicted gas pressures with no outages were slightly lowerthan any of those with outages but the difference was minimaland given more time it is likely that all the predicted pressureswould be the same.

In contrast, under ‘worst case’ conditions (Fig. 13c) the gas pres-sures predicted with no outages were higher than any of thosewith outages. This is because the occurrence of an outage, at what-ever time, interrupts the otherwise continual rise in pressure pre-dicted under ‘worst case’ conditions. Without outages the gaspressure effectively rises more quickly over the life time of thePCPV. Although still relatively small this difference could be signif-icant if the pressures approached the tensile strength of the con-crete. In that sense, although timing has little effect, any outagesare beneficial to long term structural integrity.

Considering the moisture content, under typical conditions,drying occurs throughout the life time of the structure irrespectiveof the timing of the outages and does not reach steady state evenafter 30+ years (Fig. 13b). It is maximised where no outages occurto interrupt the process and minimised by later outages whererecovery of moisture content occurs during cooling of the concrete.Under ‘worst case’ conditions (Fig. 13d), the outages have little ef-fect and very little drying occurs over the life time of the structure.

4.3.2. Longer outagesThe length of any particular outage will normally be governed

by the reason for that outage; often planned maintenance.Fig. 13e shows the pressures predicted for extended outages undertypical conditions and in comparison with the previous analyses itmay be seen that the post outage peaks in pressure were higher.However, the pressure still returns to steady conditions given suf-ficient time and over the whole life time no difference was seen.

As above, the outages caused an interruption to the drying pro-cess and so longer outages resulted in a higher ‘final’ saturation(Fig. 13f), although, given time it is likely that drying would reachthe same level as with shorter outages.

Fig. 13g shows the gas pressures predicted for extended outagesunder ‘worst case’ conditions. In this case the interruption in thegas pressure increase, discussed above, was extended and thishad the effect of reducing the ultimate gas pressure even further.Thus longer outages, although not desirable from an economic,power generation point of view, may be better for the structurein the long term.

As before, longer outages had very little effect on the drying ofthe concrete with very little moisture lost over all (Fig. 13h).

4.3.3. Drier environmentThe environmental conditions outside the PCPV clearly have an

effect on heat and mass transfer out of the concrete. While typicalconditions are reasonably well known and have been applied inmost of the analyses in this work, it is feasible to control the atmo-spheric conditions surrounding the vessel and enforce a drier envi-ronment. (Other countries may of course have naturally drierenvironments.) Here the relative humidity outside the PCPV wasreduced from 70% to 50%.

Intuitively, a drier environment will promote transport of va-pour out of the concrete, increasing the rate of evaporation insidethe concrete (drying) and result in lower gas pressures and lowersaturation. As can be seen in Fig. 13i and j, under typical conditionsthis is the case. The relatively high permeability allowed the va-pour to escape rapidly as it was driven towards the outside ofthe structure by an increased drying gradient (even before firstheat up). This reduced the gas pressure at the steel liner signifi-cantly and therefore reduced the likelihood of damage.

However, as can be seen in Fig. 13k and l, under ‘worst case’conditions the opposite is true. The increased drying gradientdrove vapour more quickly towards the outside of the structurebut the low permeability did not allow it to escape the structureas quickly as under typical conditions and almost no drying

Fig. 12. Results of analyses with ‘worst case’ combination of porosity, moisture content, permeability and thermal conductivity showing (a) gas pressures at the innerboundary in time and (b) the liquid water saturation at the inner boundary in time.



Fig. 13. Results showing gas pressures and the liquid water saturation at the inner boundary in time (a and b) with various outage timings under original conditions, (c and d)with various outage timings under ‘worst case’ conditions, (e and f) with longer outages under original conditions, (g and h) with longer outages under ‘worst case’ conditions,(i and j) with drier environmental conditions under original conditions, (k and l) with drier environmental conditions under ‘worst case’ conditions, (m and n) for extendedoperation under original and ‘worst case’ conditions.



occurred. Instead the concrete fills with more vapour and the effectis to increase the saturation slightly and the gas pressure by a sig-nificant amount at the steel liner. Thus, attempting to dry the con-crete of the PCPV by controlling the environmental conditionscould theoretically induce damage in the structure.

4.3.4. Continued operation – life extensionOne of the possible solutions to the requirement for increased

energy demand in the UK and elsewhere is to extend the life ofexisting nuclear power plants. Given the findings of the analysespreviously discussed in this work, two analyses were run to exam-ine the behaviour of the PCPV if it were to continue operating. Toillustrate the behaviour in the extreme an extended life time of200 years was analysed. As can be seen in Fig. 13m and n, undertypical conditions, the gas pressures reached equilibrium after lessthan 50 years of operation with the saturation following afteraround 70 years. Thereafter a constant relatively low level of pres-sure and saturation were maintained. However, under ‘worst case’conditions it can be seen that, while drying was occurring at a slow

rate, the gas pressure continued to rise. Although the trend waslevelling it had not reached equilibrium even after 200 years andthe pressure was approaching 2 MPa. This, in conjunction withthe findings of the first analysis (Figs. 5–7), suggests that theremay be limitations to the capacity for life extension of existingPCPVs and that before significant life extension can be considereddetailed investigation and analysis should be carried out, takinginto account the key material properties and operating conditionsover the life time of the structure, in order to provide a rationalassessment.

4.4. Temperature excursions

A further concern for operators is the possibility of temperatureexcursions, similar to those experienced at the Fukushima Dai-ichipower plant in Japan, whereby, due to a full or partial loss in cool-ing, the temperature in the PCPV can rise to temperatures perhapsin the region of 250 �C.

Fig. 13 (continued)



To examine the effects of such temperature excursions at differ-ent points during the life time of the PCPV in the context of theanalyses previously presented and by way of demonstrating thecapabilities, versatility and potential of the numerical tool em-ployed here, nine separate analyses have been conducted. The firstsix, conducted using the simplified slice model, as before considerthe gas pressures developed at the inside surface of the structureunder typical and ‘worst case’ conditions. The second three, con-ducted using the full axi-symmetric model, consider the mechani-cal response and in particular the mechanical damage thatdevelops under typical conditions. In each case it is assumed thatthe reactor is shut down following the incident and undergoes anoutage before being restarted to normal operation.

4.4.1. Simplified slice analysisAs can be seen in Fig. 14 and as may be expected, during all six

excursions, at early life, mid-life and late life, a significant spikedeveloped in the gas pressure corresponding to the increase intemperature. Under typical conditions (Fig. 14a) these spikes ex-ceeded 1.2 MPa in all cases and 1.4 MPa during the early-life excur-sion where more water remained in the concrete, but generallydecrease with later life. Under ‘worst case’ conditions these spikesincreased with later life and ultimately exceeded 3.0 MPa(Fig. 14c). As noted previously, pressures of these magnitudes arelikely to result in damage to the concrete or steel liner.

It may be further noted that the behaviour of the structurechanged following the excursions. In the case of typical conditions,rather than returning to the pre-excursion levels the gas pressuresrecovered to much lower levels. This is because the permeability ofthe concrete increased as a result of the thermal damage that the

concrete suffered during the excursion; i.e. degradation of thematerial due to heating. Similar damage occurred under ‘worstcase’ conditions but, because of the initially very low permeability,rather than dropping, the gas pressures continued to rise but at adecreased rate. It is also noticeable in this case that, during the out-ages following the temperature excursions, the gas pressures didnot drop by much and in some cases rose. This is again a functionof the low permeability. During the excursion a lot of vapour wasproduced by evaporation and this was not easily dissipated evenwhen the structure cooled during outages. The pressure increaseduring outages was a result of the flow of vapour changing direc-tion and flowing back towards the steel liner as the pressure gradi-ents changed.

The spikes in gas pressure were accompanied by changes to thesaturation (Fig. 14b and d). Under typical conditions there are sig-nificant drops in the moisture content as water is evaporated andexpelled from the structure under high temperature gradients,aided by high permeabilities in the thermally damaged concrete.Under ‘worst case’ conditions the changes are less pronouncedbut increases in the drying rates can be observed due to the in-creased (although still relatively low) permeability of the postexcursion, damaged concrete.

4.4.2. HTM analysis of PCPVExamining the results of the full analyses (Fig. 15) it can be seen

that, in all three cases, large zones of mechanical damage devel-oped on the outside of the PCPV. As before these were a result oftensile stresses caused by the outward bending of the structure un-der the influence of internally applied pressures and restrainedthermal expansion. The damaged areas were far more extensive

Fig. 14. Results of analyses of temperature excursions at early life, mid-life and late life showing gas pressures and the liquid water saturation at the inner boundary in time(a and b) with original conditions, (c and d) with ‘worst case’ conditions.



than those seen under original conditions (Fig. 4g), stretching overmost of the height of the structure and up to 3 m into the wall.Other areas of the structures, around the top and base, were alsoseen to be damaged and it is notable that, while the large damagezone to the outside of the structure is similar after all three excur-sions, the zone of damage at the base is seen to increase in size asthe excursions occur increasingly later in the life of the structure.

As a result of the significant softening of the damaged concretethe deformations of the structure were also greater. Whereas themaximum lateral deformation under typical conditions wasaround 1.3 mm, deformations under early-, mid- and late-lifeexcursions were around 6.7, 7.2 and 7.4 mm respectively. As be-fore, the increase in deformation with the later-life excursion canbe attributed to creep in the pre-stressing tendons combined withthe effects of transient thermal creep in the concrete.

The moisture content of the concrete is also affected by thetemperature excursions although it is similarly influenced in allthree stages of life. Rapid drying takes place near the hot interiorof the structure and a ring of increased saturation develops fromthe middle outwards as vapour from the centre is driven out byhigh temperature gradients and subsequently re-condenses inthe cooler parts of the concrete. This effect is reduced in the dam-aged areas where permeability in higher, allowing the vapour todissipate.

These results, in combination with the thermal damage causedby the high temperatures to which the concrete were exposed,indicate a potentially significant threat to the structural integrityof the PCPVs and clearly again, further detailed analysis shouldbe carried out in order to provide a rational assessment.

However, it must be emphasised that this is an extreme andsimplified analysis, representative of a total and uniform loss ofcooling. Other specific excursions may be more localised and lessextreme but nonetheless these analyses again serve to illustratethe capabilities of the model in predicting the fully coupled,long-term, HTM behaviour of such structures.

5. Conclusions

This work has shown the modelling of the hygro-thermo-mechanical behaviour of a typical PCPV, generically similar to

several currently in operation in the UK, over its full 30+ year lifetime. Through extensive parametric and sensitivity studies it hasbeen shown that an understanding of the long-term behaviour ofthese safety–critical structures in response to variations in materialproperties and loading conditions is extremely important if currentstructures are to be considered for life extension and new builddesigns are to be optimised. The capabilities of the fully coupled,HTM model have been demonstrated and the following conclusionsin relation to the current state of existing structures and the designand operation of new build structures may be drawn:

� Analysis of the typical structure over its full life timeshowed that some damage had gradually developed dueto outwards bending of the walls under normal operatingconditions. This was largely a result of the changes to theconcrete and the diminishing pre-stress loads and wasnot considered to be significant in terms of the structuralintegrity or safety of the structure at present. However,the findings suggest that further detailed analysis isrequired in order to provide a rational assessment for lifeextension.

� The parametric study considering material properties thatare typically less well characterised showed that under cer-tain conditions gas pressures in the concrete (which maycontribute adversely to the stress state in the structure,ultimately leading to the development of damage) maycontinue to increase over the whole life-time of the struc-ture, representing a worsening case over the operating lifetime. It is feasible that these pressures may exceed 1 MPa.

Permeability, which may be the least well characterised prop-erty of the concrete, was found to have the largest and most signif-icant effect on the gas pressures, while the porosity, moisturecontent and thermal conductivity were found have more limitedand respectively decreasing effects. The ‘worst case’ conditions,potentially leading to gas pressures of over 1.3 MPa, were foundto be concrete of low permeability, high porosity, high moisturecontent and low thermal conductivity. These factors and the con-trol of these properties should be considered when designingnew build structures.

Fig. 15. Deformation of structure at an exaggerated scale (�450) showing mechanical damage and liquid water saturation resulting from temperature excursions at (a and b)early life, (c and d) mid-life, (e and f) late-life.



� The sensitivity study found that only minor changes wereseen in the development of gas pressures as a result ofchanges to the operating procedures. Outages of any lengthhad minimal effect at low pressures but acted to interruptand delay the development of persistently increasing gaspressures. In that sense, outages were found to be benefi-cial for the long term prevention of damage. Changes tothe environmental conditions outside the PCPV had thepotential to beneficially or adversely affect the behaviourdepending on the properties of the concrete.

� Temperature excursions, as may occur as a result of a lossof cooling, were found to potentially have very significanteffects for the PCPV. Under analyses representing veryextreme cases it was found that very high gas pressurescould develop within the concrete potentially damagingthe structure and the steel liner. Extensive thermal andmechanical damage were found to occur in the concretewalls of the structure affecting its strength and stiffnessand certainly requiring further detailed analysis in orderto provide a rational assessment for continued operation.

Acknowledgements

This work was carried out with the aid of funding from theEuropean Community through the MAECENAS research project‘‘Modelling of ageing in concrete nuclear power plants’’ (FIKS-CT-2001-00186) and UK Engineering and Physical Sciences ResearchCouncil research project ‘‘An Advanced Numerical Tool for the Pre-diction and Analysis of Spalling in Concrete Structures Exposed toCombined Thermal and Mechanical Loading’’ (EP/E048935/1). Thesignificant contribution of Mr Adam Gibson is also gratefullyacknowledged.

Appendix A

Full details of the parametric relationships employed in themodel can be found in [6,7,17]. Brief details for some key parame-ters are given here.

Intrinsic permeability:

K ¼ K0ð104DÞ ðAI:1Þ

where K0 is the initial permeability, and D = x + v �xv is the mul-tiplicative thermal (v)-mechanical (x) damage.

Thermal conductivity [19]:

k ¼ k1 � k2TC

100

� �� k3

TC

100

� �2

ðAI:2Þ

where the coefficients k1 = 1.68, k2 = 0.19055 and k3 = 0.0082.Moisture content – Sorption Isotherms (8) (adapted from [21]):

eL ¼

eCemqCemqL

� �u0q0

LeCemqCem

PVPSat

� �1=mfor PV

PSat

� �6 0:96

a PVPSat

� �3þ b PV

PSat

� �2þ c PV

PSat

� �þ d for 0:96 < PV

PSat

� �< 1:00

u for PVPSat

� �¼ 1:00

8>>>>><>>>>>:ðAI:3Þ

where /0 is the initial porosity of the concrete, rho0L is the initial

density of liquid water, a, b, c and d are complex temperaturedependent coefficients of a cubic function such that eL and its deriv-atives are always continuous, and m is a temperature dependentcoefficient given by (AI.4):

m ¼ 1:04� ðTC þ 10Þ2

ðTC þ 10Þ2 þ 22:3ð25þ 10Þ2ðAI:4Þ

where TC is the temperature in degrees Celsius.Coefficient of Load Induced Thermal Strain (13):

b ¼ 0:01� 2Xhþ Y for 0 6 h 6 h

2Zðh� hÞ þ 2Xhþ Y for h > h

(ðAI:5Þ

where X, Y and Z are coefficients of a bi-parabolic function.Tensile strength:

ft ¼ f 0t ð1� 0:1bhÞ2 ðAI:6Þ

where f 0t is the initial tensile strength of the concrete andbh ¼ maxðhÞ, where h is a normalised temperature defined by:

h ¼ T � T0

100ðAI:7Þ

References

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